Simplify Propositional Form I was solving this problem.
Simplify propositional form:
 $$[(p\implies q) \vee (p \implies r)] \implies (q \vee r)$$
I was using this fact for solving:
$$(p \implies q) \equiv (\neg p \vee q)$$
but end up with
$$(p \implies (q \vee r)) \implies (q \vee r)$$
I need help to further simplifying this expression. Any kind of help would be appreciated. Thanks!
 A: HINT. If looking for a simpler expression:
$$
((p\implies q) \lor (p \implies r)) \implies (q \lor r) \\
 \equiv  \lnot((\lnot p \lor q) \lor (\lnot p \lor r)) \lor (q \lor r) \\
 \equiv  \lnot(\lnot p \lor q \lor r) \lor (q \lor r) \\
 \equiv  ( p \land \lnot (q \lor r)) \lor (q \lor r)\\
 \equiv  ( p \lor q \lor r ) \land (\lnot (q \lor r) \lor (q \lor r)) \\
 \equiv  p \lor q \lor r  
$$
A: $(A \implies B) \equiv \neg A\lor B$
In you expression 
$A$ is $(p\implies q) \vee (p \implies r)$ and $B$ is $q \vee r$
Thus the simplification is
$(p\Rightarrow q)\lor (p\Rightarrow r)\Rightarrow q\lor r \equiv \neg ((\neg p\lor q)\lor (\neg p\lor r))\lor (q\lor r)$
which simplifies to
$p\lor q\lor r$
Hope this helps
A: $$[(p\implies q) \vee (p \implies r)] \implies (q \vee r) \equiv \text{ (Implication)}$$
$$\neg [(\neg p \lor q) \vee (\neg p \lor r)] \lor (q \vee r) \equiv \text{ (Association)}$$
$$\neg (\neg p \lor q \vee \neg p \lor r) \lor q \vee r \equiv \text{ (Idempotence)}$$
$$\neg (\neg p \lor q \lor r) \lor q \vee r \equiv \text{ (DeMorgan)}$$
$$(p \land \neg q \land \neg r) \lor q \vee r \equiv \text{ (Reduction)}$$
$$p \lor q \vee r$$
At the end I used:
Reduction
$P \land (\neg P \lor Q) \Leftrightarrow P \land Q$
$P \lor (\neg P \land Q) \Leftrightarrow P \lor Q$
